Calculate the derivative of \( f(x) = 3x^3 - 5x^2 + 2x - 7 \).
["# How to Calculate the Derivative of ( f(x) = 3x^3 - 5x^2 + 2x - 7 ) – A Complete Guide", "Calculating the derivative of a function is a fundamental concept in calculus, essential for understanding rates of change, optimization, and curve analysis. In this article, we’ll explore how to carefully compute the derivative of the polynomial function:", "[\nf(x) = 3x^3 - 5x^2 + 2x - 7\n]", "Understanding the differentiation process helps students, engineers, and analysts unlock deeper mathematical insights. Let’s break down the steps.", "---", "### What is a Derivative?", "The derivative of a function at a point gives the slope of the tangent line to the curve at that point. For polynomial functions, the derivative follows simple, predictable rules—knowing these makes differentiation efficient and intuitive.", "---", "### Step-by-Step Derivation", "Given the function:", "[\nf(x) = 3x^3 - 5x^2 + 2x - 7\n]", "We apply the power rule of differentiation, which states that the derivative of ( x^n ) is ( nx^{n-1} ).", "Step 1: Differentiate each term individually", "1. Derivative of ( 3x^3 ):\n ( 3 \cdot 3x^{3-1} = 9x^2 )", "2. Derivative of ( -5x^2 ):\n ( -5 \cdot 2x^{2-1} = -10x )", "3. Derivative of ( 2x ):\n ( 2 \cdot 1x^{1-1} = 2 )", "4. Derivative of constant ( -7 ):\n The derivative of any constant is 0.", "Step 2: Combine all terms", "Add the derivatives of each term:", "[\nf'(x) = 9x^2 - 10x + 2 + 0\n]", "So, the final derivative is:", "[\nf'(x) = 9x^2 - 10x + 2\n]", "---", "### Why This Matters", "This result tells you everything you need to know about how ( f(x) ) behaves:", "- The local maxima and minima (critical points) occur where ( f'(x) = 0 ).\n- The rate at which ( f(x) ) is increasing or decreasing is given by the sign and magnitude of ( f'(x) ).\n- This derivative is used in physics for velocity calculations, in economics for marginal cost, and in optimization problems across disciplines.", "---", "### Summary", "To calculate the derivative of ( f(x) = 3x^3 - 5x^2 + 2x - 7 ), apply the power rule term-by-term:", "[\n\boxed{f'(x) = 9x^2 - 10x + 2}\n]", "Learning to differentiate polynomial functions builds a strong foundation in calculus and enhances problem-solving skills in STEM fields.", "---", "Keywords for SEO: derivative calculation, how to find f’(x), differentiate polynomial function, power rule derivative, calculus tutorial, f'(x) example, polynomial derivative, calculus step-by-step", "Always remember: mastery of derivatives starts with practice and understanding fundamental rules like the power rule. Dive into more functions, explore the chain rule, and watch your calculus skills grow!"]
